Duval, Mylene (2008) Modelisation and estimation of heterogeneous variances in nonlinear mixed models. PhD thesis Mathématiques appliquées , INRA - SAGA, AgroParistech 2008AGPT0082 p.186.
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Abstract
Nonlinear mixed models stand apart in mixed models methodology. Contrary to linear
and generalized linear models, often used as black boxes, the trajectory function t in
nonlinear models generally comes from the integration of dierential equations. This
provides a biological interpretation for the parameters, whereas models are often more
parsimonious. However, the estimation of the parameters in nonlinear mixed models is
complex because random eects cannot be integrated out of the likelihood in closed form.
As in all mixed models, especially those used to analyze longitudinal data, nonlinear models
are well adapted to take into account between and within-cluster variation. However,
one of the common assumptions of such models is that of independent, identically distributed
residuals with a common variance; this assumption is unrealistic in many elds
of applications.
The objective of this study was to propose some models for the residual variance to take
into account the potential heterogeneity of variance of the residuals, while limiting the
number of parameters in these models. In this sense, we used a parametric approach
based on a linear mixed model on the logvariance, as well as the classical power \meanvariance"
function.
A classical inference method based on maximum likelihood theory was selected and we
considered a stochastic EM algorithm, the SAEM-MCMC algorithm. The mixed model
structure applied to the position and dispersion parameters is well adapted to the implementation
of EM algorithms. Some instrumental distributions adapted to the analysis
of these models, as well as some convergence criteria, were proposed in the MCMC step.
The overall algorithm was numerically validated in both linear and nonlinear models, by
comparing its results with those of an analytical EM algorithm (in the linear case) or
other algorithms like those based on Gaussian quadrature.
Finally, an application to the analysis of somatic cell scores in dairy cattle was presented.
Several linear and nonlinear models were compared showing a clear gain obtained taking
into account the heterogeneity of variances.
| Item Type: | PhD Thesis (PhD) |
|---|---|
| PhD Supervisor: | Foulley, Jean-Louis and Robert-Granié, Christèle |
| Date: | 08 December 2008 |
| Board of examiners: | Verrier, Etienne and Concordet, Didier and Parent, Eric and Samson, Adeline |
| Ecole Doctorale: | ED 435 AGRICULTURE, ALIMENTATION, BIOLOGIE, ENVIRONNEMENTS ET SANTE |
| Discipline: | Mathématiques appliquées |
| Collection (Fonds): | AgroParistech |
| Institution: | AgroParistech |
| Department: | INRA - SAGA |
| Subjects: | 1. Mathematics and Applications |
| Uncontrolled Keywords: | Modèles mixtes non linéaires, Hétéroscédasticité, Vraisemblance, algorithme SAEM, Nonlinear mixed models, Heteroskedasticity, Ikelihood, SAEM algorithm |
| ID Code: | 4846 |
| Deposited By: | Mylene Duval |
| Deposited On: | 09 April 2009 |
Table of content
1 Modèles non linéaires mixtes et variances hétérogènes : éléments bibliographiques 21
1.1 Les modèles non linéaires mixtes - 21
1.1.1 Les applications - 21
1.1.2 Les modèles - 26
1.2 Variances hétérogènes : existence et modélisation - 29
1.2.1 Existence des variances hétérogènes - 29
1.2.2 Modélisation des variances résiduelles - 31
1.3 Méthodes d'estimation dans les modèles non linéaires à effets mixtes - 34
1.3.1 Méthodes de linéarisation de la vraisemblance - 35
1.3.2 Méthodes basées sur la théorie du maximum de vraisemblance . . . 37
1.3.3 Algorithme basée sur une pseudo vraisemblance - 42
1.3.4 Méthodes bayésiennes - 42
2 Quelques critéres pour calibrer les paramêtres de l'algorithme SAEM-
MCMC 45
2.1 Introduction - 48
2.2 The nonlinear mixed effects model and the SAEM-MCMC algorithm . . . 48
2.2.1 The model - 48
2.2.2 The SAEM-MCMC algorithm - 49
2.2.3 The Metropolis-Hastings algorithm - 51
2.2.4 Estimations of the log-likelihood and standard errors - 51
2.3 The criteria - 52
2.4 Application and simulation - 54
2.5 Discussion - Conclusion - 57
2.6 Acknowledgment - 58
2.7 References - 58
3 Modélisation et estimation des variances hétérogènes dans les modèles
non linéaires mixtes 67
3.1 Introduction - 70
3.2 The heteroskedastic nonlinear mixed Model - 71
3.3 A monitored SAEM-MCMC algorithm - 72
3.3.1 Computation of the ML estimations - 72
3.3.2 Estimations of the log-likelihood and standard errors - 74
3.4 Numerical applications - 75
3.4.1 Validation on a linear model: Potho and Roy's data - 75
3.4.2 Application to a non linear mixed model : growth curves in poultry 76
3.5 Discussion - Conclusion - 78
3.6 References - 80
3.7 Appendix - 82
3.7.1 The Metropolis-Hastings algorithm - 82
3.7.2 Some criteria to calibrate the parameters of the SAEM-MCMC algorithm
- 82
3.7.3 Parameters of the SAEM-MCMC algorithm used in Potho and
Roy's data analysis - 83
4 Application : modéliser les cinétiques de scores de cellules somatiques
chez les bovins laitiers 91
4.1 Introduction - 91
4.2 Matériel et Méthodes - 93
4.2.1 Les données - 93
4.2.2 Le modèle statistique - 93
4.2.3 Méthode d'estimation - 96
4.2.4 Sélection de modèles - 97
4.3 Résultats - 100
4.3.1 Etape 1 : Choix des meilleurs modèles linéaire et non linéaire homogènes - 100
4.3.2 Etape 2 : Déterminer les effets fixes et aléatoires de la fonction
moyenne - 101
4.3.3 Etape 3 : Sélection des covariables sur le vecteur moyenne - 102
4.3.4 Etape 4 : Choix du modèle de variance résiduelle - 103
4.3.5 Etapes 5 : Choix des covariables pour les paramêtres des fonctions
de variances [V4] et [V5] - 104
4.4 Discussion et conclusion - 105
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